Question

Problem 7: 15 points. For any a ER, define the binomial-type coefficient by a(a – 1)(a − 2)... (a – n +1) C(a, n) = n! for n > 1 and C(a,0) = 1. For example, C(1,2)= Let g(x) = {nco C(2, n)x". (a) Determine the radius of convergence r for the power series g(x). (b) Determine the power series for the derivative g(x) on the interval (-r,r). (c) Show : g(x) = 2-0 C(,n)z” by examining the coefficients in part (b). 2 2n=0

Answer 1

solution Clam) = a caro ca-2)...carneol for n> and ċ (0,0)=1 - 96x= E CCğın = C(1,0) + C($11) x + c(+12)x2 toru. as = 1+ {x + (x2+.... g(x) = (1+x)%3 To find radius of convergence. we have L= limpos 1 amor = lim 1241 a Ca-1) ... ( a-h-1+)) n100 n+D! a cari... Ca-nti = lain I ~ I am in CS$canned with CamScanner

nto L = lim imet le content - Irel line o 192 ė irl .1-11 = 1 for convergence L < 1 = 1 x < 1 Hence radius of convergence IS R=1 ② power seues of gx) is given by g(x) = { c (2) g() = c(tim). nent = ECC £; n+1) 6+1) se? © How to prove that Cena nti), (n+1) = { c(tim) ? C(2,) = -Ź 64-1)(*_)... (3-+1) Icetn) = (3) (+) (4-564-3)...-(-) n! ree C. Scared with Capascanner

#c(--) = \(\-)(\-) --+-++)+1) (+) (+))! = C(£,n+i). (n+1) Hence 700) = h cho nt) Chr) en : = Į & CC-+,n) x? Hence proved CSScanned with CamScanne-